1. **Problem statement:** A circle is inscribed inside a square such that the circle touches all four sides of the square. Given the area of the circle is 139.7 cm², find the area of the square. 2. **Formula for the area of a circle:** $$A_{circle} = \pi r^2$$ where $r$ is the radius of the circle. 3. **Find the radius of the circle:** Given $A_{circle} = 139.7$, we solve for $r$: $$r = \sqrt{\frac{A_{circle}}{\pi}} = \sqrt{\frac{139.7}{3.1416}}$$ 4. Calculate the radius: $$r \approx \sqrt{44.45} \approx 6.67 \text{ cm}$$ 5. **Relation between the circle and the square:** Since the circle touches all four sides of the square, the diameter of the circle equals the side length of the square. 6. Calculate the side length of the square: $$s = 2r = 2 \times 6.67 = 13.34 \text{ cm}$$ 7. **Calculate the area of the square:** $$A_{square} = s^2 = (13.34)^2 = 177.96 \text{ cm}^2$$ **Final answer:** The area of the square is approximately $177.96$ cm².